Detection of gravitational waves and the expansion of the universe Very recently, there was the fascinating news of new measurements being done 
 by LIGO, in this case detecting the gravitational waves of two Neutron stars colliding. These detections prompted the astronomers to also look for the emitted electromagnetic waves of the aftermath, given that now they knew in which direction to look for them thanks to the GW measurements. 
These discoveries make for really exciting times in physics, I personally get goosebumps when I read about them. Now admittedly, it is very difficult to grasp the physics that is going on here for someone not involved in the field, but it would be incredibly valuable if, briefly, some light could be shed on how these new discoveries relate to the problem of rate of expansion of the universe, potentially enabling physicists to further test the Hubble number. Such that one can have a rough idea at least at a conceptual level what the key idea behind this relation may be.
More concretely:


*

*As a small preliminary question, in our current understanding, what are the main elements that play a role in the balance between the expansion and contraction of the universe? What one often hears, is that there's a strong interplay between how matter, EM radiation and potentially dark energy influence the rate of expansion of the univserse. 

*How does the recent detection of GW's connect to the study of expansion rate of the universe?  
This is merely an attempt to get more input in order to understand some of the main ideas involved, as this is very exciting news given the long history behind these problems and predictions, and all the debates around them.
 A: *

*Let's just make a quick stop at each of the key ingredients:


*

*Matter (ordinary one and dark one) slows the expansion through gravitational attraction. As the time passes, density of matter decreases, 'cuz there's only so much matter and progressively more and more space.

*Dark energy is assumed to be the energy of space itself and has a constant density: the same volume of space always has the same dark energy. It accelerates the expansion through gravitational "repulsion".

*Radiation can, in principle, accelerate the expansion through its pressure, and that was the case in the early universe. However, nowadays it's effect is next to none.
Note that as the space expands, quantity of dark energy increases, so as time passes, we have more and more contribution from dark energy. About 6 billion years ago (pls correct me on exact numbers) contribution of dark energy became greater than the one of matter, so expansion of the universe is now accelerating.


*The main point of detection GW was not that it will help us to understand the expansion, but more that it's the direct observational evidence of general relativity (however, the accuracy is not yet enough to choose between different variatons of the theory). However, the latest registered event (from the merger of two NS) already gave us a new insight into the nature of gamma-ray bursts, for example.
As for the last part of your question, the one possible usage I can think out is a new kind of standard candles. There are some calculations showing that the luminosity of kilonova (that's how they call an optical transient of the merger) should have pretty standard values, and so the luminosity data can be used to calculate distances.
I'm sure there are many more ways to use those observations, however I'm still quite new to the field.
A: Part 1 of the question is already answered quite well, so not going to dwell on that.
Part 2 is:


*

*The LIGO experiment has assured us is that we can actually detect the gravitational waves because this time not only we have detected them on LIGO, we literally saw the event from other instruments.

*The new discovery helped us realize that the gamma bursts don't work the way we thought they did and need improvements.

*It will help us map dense collisions and if there are effects of those collisions then we might look into it.

*To really answer your question on what this will help us in: we don't know. Even Einstein didn't know the potential outcomes of his Nobel-prize-winning paper.


But until we understand dark energy and dark matter, we can't really come up with a precise expansion rate.
A: Actually, the main cosmological contribution from the kilonova binary neutron star (BNS) detection was an independent measurement of the Hubble constant, $H_0$. It came back as 70 km/(sec Mpsec), with uncertianties (68%) of +- 10 approx. The accuracies will improve over time with higher SNRs and a greater number of gravitational wave detections.
The independence is due to the distance-ladder independent measurement of distance using the gravitational wave (GW) parameters and analysis as a standard siren. Using electromagnetic redshifts for the Galaxy, and doing some peculiar velocity adjustments, the Hubble constant is then obtained. Within the accuracy limits it is consistent with Planck CMB measurements. See the LIGO papers, at http://www.nature.com/nature/journal/vaap/ncurrent/pdf/nature24471.pdf and at https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.119.161101
In the longer term, probably in the 2020s space based gravitational observatories with longer arm lengths (currently eLISA is proposed to have about 1 million Kms arm lengths) will allow observations of larger scale and longer wavelength structures such as possible anomalies early after the Big Bang. With those and improved earth based detectors we will be able to see beyond the current electromagnetic wall around tHe radiation decoupling at about 300,000 light years after the BIg Bang. We will be seeing much more energetic physics, and look for primordial structures like branes, or domain walls, or primordial black holes, as well as possible detections of GW waves from inflation, and other exotic phenomena, we will have a chance to probe those scales with GW detections. See eg https://arxiv.org/pdf/1405.0504.pdf
GWs open up a whole new window of observations, a completely new and independent messenger of information to add to all our electromagnetic observations.  
